Go back to the Convergence Rates page.

Let us set some global options for all code chunks in this document.

knitr::opts_chunk$set(
  message = FALSE,    # Disable messages printed by R code chunks
  warning = FALSE,    # Disable warnings printed by R code chunks
  echo = TRUE,        # Show R code within code chunks in output
  include = TRUE,     # Include both R code and its results in output
  eval = TRUE,       # Evaluate R code chunks
  cache = FALSE,       # Enable caching of R code chunks for faster rendering
  fig.align = "center",
  #out.width = "100%",
  retina = 2,
  error = TRUE,
  collapse = FALSE
)
rm(list = ls())
set.seed(1982)

1 Import libraries

# Install R-INLA package
# install.packages("INLA",repos = c(getOption("repos"),INLA ="https://inla.r-inla-download.org/R/testing"), dep = TRUE)
# Update R-INLA package
# inla.upgrade(testing = TRUE)
# Install inlabru package
# remotes::install_github("inlabru-org/inlabru", ref = "devel")
# Install rSPDE package
# remotes::install_github("davidbolin/rspde", ref = "devel")
# Install MetricGraph package
# remotes::install_github("davidbolin/metricgraph", ref = "devel")

library(INLA)
library(inlabru)
library(rSPDE)
library(MetricGraph)
library(Matrix)

library(dplyr)
library(plotly)
library(scales)
library(patchwork)
library(tidyr)
library(ggplot2)
library(reshape2)
library(sf)

library(here)
library(rmarkdown)
library(knitr)
library(grateful) # Cite all loaded packages

library(latex2exp)
library(plotrix)

rm(list = ls()) # Clear the workspace
set.seed(1982) # Set seed for reproducibility

2 Tadpole graph

2.1 Define utility functions

# Eigenfunctions for the tadpole graph
tadpole.eig <- function(k,graph){
  x1 <- c(0,graph$get_edge_lengths()[1]*graph$mesh$PtE[graph$mesh$PtE[,1]==1,2]) 
  x2 <- c(0,graph$get_edge_lengths()[2]*graph$mesh$PtE[graph$mesh$PtE[,1]==2,2]) 
  
  if(k==0){ 
    f.e1 <- rep(1,length(x1)) 
    f.e2 <- rep(1,length(x2)) 
    f1 = c(f.e1[1],f.e2[1],f.e1[-1], f.e2[-1]) 
    f = list(phi=f1/sqrt(3)) 
    
  } else {
    f.e1 <- -2*sin(pi*k*1/2)*cos(pi*k*x1/2) 
    f.e2 <- sin(pi*k*x2/2)                  
    
    f1 = c(f.e1[1],f.e2[1],f.e1[-1], f.e2[-1]) 
    
    if((k %% 2)==1){ 
      f = list(phi=f1/sqrt(3)) 
    } else { 
      f.e1 <- (-1)^{k/2}*cos(pi*k*x1/2)
      f.e2 <- cos(pi*k*x2/2)
      f2 = c(f.e1[1],f.e2[1],f.e1[-1],f.e2[-1]) 
      f <- list(phi=f1,psi=f2/sqrt(3/2))
    }
  }
  
  return(f)
}


# Function to compute the true covariance matrix
gets_true_cov_mat <- function(graph, kappa, tau, alpha, n.overkill){
  Sigma.kl <- matrix(0,nrow = dim(graph$mesh$V)[1],ncol = dim(graph$mesh$V)[1])
  for(i in 0:n.overkill){
    phi <- tadpole.eig(i,graph)$phi
    Sigma.kl <- Sigma.kl + (1/(kappa^2 + (i*pi/2)^2)^(alpha))*phi%*%t(phi)
    if(i>0 && (i %% 2)==0){ 
      psi <- tadpole.eig(i,graph)$psi
      Sigma.kl <- Sigma.kl + (1/(kappa^2 + (i*pi/2)^2)^(alpha))*psi%*%t(psi)
    }
    
  }
  Sigma.kl <- Sigma.kl/tau^2
  return(Sigma.kl)
}

2.2 Define the graph

edge1 <- rbind(c(0,0), c(1,0))
theta <- seq(from = -pi, to = pi,length.out = 100)
edge2 <- cbind(1+1/pi+cos(theta)/pi, sin(theta)/pi)
edges <- list(edge1, edge2)
graph <- metric_graph$new(edges = edges)
graph$plot() + 
  ggtitle("Tadpole graph") + 
  theme_minimal() + 
  theme(text = element_text(family = "Palatino")) +
  coord_fixed(ratio = 1)

2.3 Define the parameters

# parameters
h.ok <- 2^-9
type <- "covariance"
type_rational_approximation = "brasil"
rho <- 0.5
#m = 4
sigma <- 1
n.overkill = 1000

# Mesh sizes
h_aux <- c(6:3)
h_vector <- 2^-h_aux
h_label <- paste0("2^-", h_aux, "")
h_label_latex <- sprintf("$2^{-%d}$", h_aux)

# Beta values
beta_aux <- c(4, 4.5, 5, 5.5, 6, 6.5, 7)
beta_vector <- beta_aux/8
beta_label <- paste0(beta_aux, "/8")
theoretical_rate <- pmin(4*beta_vector-1/2,2)

2.4 Build the graph with overkill mesh

graph.ok <- graph$clone()
# Build graph with overkill mesh
graph.ok$build_mesh(h = h.ok)

graph.ok$plot(mesh = TRUE) +
  ggtitle("Tadpole graph with overkill mesh") +
  theme_minimal() +
  theme(text = element_text(family = "Palatino")) +
  coord_fixed(ratio = 1)

# Get the overkill mesh locations
loc.ok <- graph.ok$mesh$VtE # or graph.ok$get_mesh_locations()

Each A[[i]] below is the projection matrix corresponding to the graph with mesh size h_vector[i] onto the overkill mesh.

# Initialize the list of graphs and the list of projection matrices
graphs <- list()
A <- list()
for(i in 1:length(h_vector)){
  graphs[[i]] <- graph$clone()
  graphs[[i]]$build_mesh(h = h_vector[i])
  A[[i]] <- graphs[[i]]$fem_basis(loc.ok)
}
# Print the dimensions of the projection matrices
print(lapply(A, dim))
## [[1]]
## [1] 1536  192
## 
## [[2]]
## [1] 1536   96
## 
## [[3]]
## [1] 1536   48
## 
## [[4]]
## [1] 1536   24

2.5 Compute the covariance error

cov.error.ok.mesh <- matrix(NA, nrow = length(h_vector), ncol = length(beta_vector))
cov.error.folded <- matrix(NA, nrow = length(h_vector), ncol = length(beta_vector))

par(family = "Palatino")
layout(matrix(1:(length(beta_vector)*length(h_vector)),
              nrow = length(beta_vector),
              byrow = TRUE))

m_values <- c()
for (j in 1:length(beta_vector)) {
  beta <- beta_vector[j]
  alpha <- 2*beta
  fract2beta <- alpha - floor(alpha)
  nu <- alpha - 0.5
  kappa <- sqrt(8*nu)/rho
  tau <- sqrt(gamma(nu) / (sigma^2 * kappa^(2*nu) * (4*pi)^(1/2) * gamma(nu + 1/2)))  #sigma = 1, d = 1

  Sigma.folded <- gets_true_cov_mat(graph = graph.ok,
                               kappa = kappa,
                               tau = tau,
                               alpha = alpha,
                               n.overkill = n.overkill)

  for (i in 1:length(h_vector)) {
    h <- h_vector[i]
    m <- min(20, ceiling((min(4*beta - 1/2,2) + 1/2)^2*log(h)^2/(4*pi^2*fract2beta)))
    m_values <- c(m_values, m)

    Sigma.ok.mesh <- matern.operators(alpha = alpha, # using overkill mesh
                                kappa = kappa,
                                tau = tau,
                                m = m,
                                graph = graph.ok,
                                type = type,
                                type_rational_approximation = type_rational_approximation)$covariance_mesh()

    Sigma <- matern.operators(alpha = alpha,  # this is the approximation
                             kappa = kappa,
                             tau = tau,
                             m = m,
                             graph = graphs[[i]],
                             type = type,
                             type_rational_approximation = type_rational_approximation)$covariance_mesh()

    Sigma.approx <- A[[i]]%*%Sigma%*%t(A[[i]])

    cov.error.ok.mesh[i,j] <- sqrt(as.double(t(graph.ok$mesh$weights)%*%(Sigma.ok.mesh - Sigma.approx)^2%*%graph.ok$mesh$weights))
    cov.error.folded[i,j] <- sqrt(as.double(t(graph.ok$mesh$weights)%*%(Sigma.folded - Sigma.approx)^2%*%graph.ok$mesh$weights))

    aux <- dim(Sigma.approx)[1]
    reordering <- c(1,3:aux,2)
    auxhalf <- reordering[length(reordering) %/% 2 + 1]

    plot(Sigma.folded[auxhalf, reordering], col = "black", type = "l", main = bquote(beta == .(beta_label[j]) ~ "," ~ h == .(h_label[i])), xlab = "", ylab = "", lwd = 2, lty = 1)
    lines(Sigma.ok.mesh[auxhalf, reordering], col = "red", lty = 2, lwd = 2)
    lines(Sigma.approx[auxhalf, reordering], col = "blue", lty = 3, lwd = 2)
    legend("topleft", lwd = 2, col = c("black", "red", "blue"), legend = c("true", "ok.mesh", "approx"), lty = c(1,2,3))
  }
}

print(m_values)
##  [1] 20 20 20 20 18 13  8  5 11  8  5  3  8  6  4  2  6  4  3  2  5  4  2  2  4
## [26]  3  2  1
# theoretical
theoretical_rate
## [1] 1.50 1.75 2.00 2.00 2.00 2.00 2.00
# folded
for (u in 1:length(beta_vector)) {print(coef(lm(log(cov.error.folded[,u]) ~ log(h_vector))))}
##   (Intercept) log(h_vector) 
##     0.1731924     1.4498299 
##   (Intercept) log(h_vector) 
##     0.4465996     1.6381700 
##   (Intercept) log(h_vector) 
##     0.6274769     1.7777247 
##   (Intercept) log(h_vector) 
##     0.6852249     1.8576150 
##   (Intercept) log(h_vector) 
##     0.6937527     1.9022237 
##   (Intercept) log(h_vector) 
##     0.6694985     1.9226518 
##   (Intercept) log(h_vector) 
##     0.5039852     1.8960870
# ok.mesh
for (u in 1:length(beta_vector)) {print(coef(lm(log(cov.error.ok.mesh[,u]) ~ log(h_vector))))}
##   (Intercept) log(h_vector) 
##     0.1592589     1.4441568 
##   (Intercept) log(h_vector) 
##     0.4506415     1.6398464 
##   (Intercept) log(h_vector) 
##     0.6679511     1.7933716 
##   (Intercept) log(h_vector) 
##     0.7861186     1.8949556 
##   (Intercept) log(h_vector) 
##     0.8361226     1.9559881 
##   (Intercept) log(h_vector) 
##     0.8411164     1.9887793 
##   (Intercept) log(h_vector) 
##      0.798770      1.998613

Press the Show button below to reveal the code.

loglog_line_equation <- function(x1, y1, slope) {
  b <- log10(y1 / (x1 ^ slope))

  function(x) {
    (x ^ slope) * (10 ^ b)
  }
}

guiding_lines <- matrix(NA, nrow = length(h_vector), ncol = length(beta_vector))
for (j in 1:length(beta_vector)) {
  guiding_lines_aux <- matrix(NA, nrow = length(h_vector), ncol = length(h_vector))
  for(k in 1:length(h_vector)){
    point_x1 <- h_vector[k]
    point_y1 <- cov.error.ok.mesh[k, j]
    slope <- theoretical_rate[j]
    line <- loglog_line_equation(x1 = point_x1, y1 = point_y1, slope = slope)
    guiding_lines_aux[,k] <- line(h_vector)
  }
  guiding_lines[,j] <- apply(guiding_lines_aux, 1, mean)
}

# Generate default ggplot2 colors
default_colors <- scales::hue_pal()(ncol(guiding_lines))

# Create the plot_lines list with different colors for each line
plot_lines <- lapply(1:ncol(guiding_lines), function(i) {
  geom_line(data = data.frame(x = h_vector, y = guiding_lines[, i]),
            aes(x = x, y = y), color = default_colors[i], linetype = "dashed", show.legend = FALSE)
})

df <- as.data.frame(cbind(h_vector, cov.error.ok.mesh))
colnames(df) <- c("h_vector", beta_label)
df_melted <- melt(df, id.vars = "h_vector", variable.name = "column", value.name = "value")

p <- ggplot() +
  geom_line(data = df_melted, aes(x = h_vector, y = value, color = column)) +
  geom_point(data = df_melted, aes(x = h_vector, y = value, color = column)) +
  plot_lines +
  labs(title = "Covariance error using overkill mesh",
    x = TeX("$\\hat{h}$"),
       y = "Covariance Error",
       color = expression(beta)) +
  scale_x_log10(breaks = h_vector, labels = TeX(h_label_latex)) +
  scale_y_log10() +
  theme_minimal() +
  theme(text = element_text(family = "Palatino"))

Press the Show button below to reveal the code.

guiding_lines <- matrix(NA, nrow = length(h_vector), ncol = length(beta_vector))
for (j in 1:length(beta_vector)) {
  guiding_lines_aux <- matrix(NA, nrow = length(h_vector), ncol = length(h_vector))
  for(k in 1:length(h_vector)){
    point_x1 <- h_vector[k]
    point_y1 <- cov.error.folded[k, j]
    slope <- theoretical_rate[j]
    line <- loglog_line_equation(x1 = point_x1, y1 = point_y1, slope = slope)
    guiding_lines_aux[,k] <- line(h_vector)
  }
  guiding_lines[,j] <- apply(guiding_lines_aux, 1, mean)
}

# Generate default ggplot2 colors
default_colors <- scales::hue_pal()(ncol(guiding_lines))

# Create the plot_lines list with different colors for each line
plot_lines <- lapply(1:ncol(guiding_lines), function(i) {
  geom_line(data = data.frame(x = h_vector, y = guiding_lines[, i]),
            aes(x = x, y = y), color = default_colors[i], linetype = "dashed", show.legend = FALSE)
})

df <- as.data.frame(cbind(h_vector, cov.error.folded))
colnames(df) <- c("h_vector", beta_label)
df_melted <- melt(df, id.vars = "h_vector", variable.name = "column", value.name = "value")

q <- ggplot() +
  geom_line(data = df_melted, aes(x = h_vector, y = value, color = column)) +
  geom_point(data = df_melted, aes(x = h_vector, y = value, color = column)) +
  plot_lines +
  labs(title = "Covariance error using folded",
    x = TeX("$\\hat{h}$"),
       y = "Covariance Error",
       color = expression(beta)) +
  scale_x_log10(breaks = h_vector, labels = TeX(h_label_latex)) +
  scale_y_log10() +
  theme_minimal() +
  theme(text = element_text(family = "Palatino"))
p + q

3 References

cite_packages(output = "paragraph", out.dir = ".")

We used R version 4.4.0 (R Core Team 2024) and the following R packages: here v. 1.0.1 (Müller 2020), htmltools v. 0.5.8.1 (Cheng et al. 2024), INLA v. 24.6.27 (Rue, Martino, and Chopin 2009; Lindgren, Rue, and Lindström 2011; Martins et al. 2013; Lindgren and Rue 2015; De Coninck et al. 2016; Rue et al. 2017; Verbosio et al. 2017; Bakka et al. 2018; Kourounis, Fuchs, and Schenk 2018), inlabru v. 2.10.1.9010 (Yuan et al. 2017; Bachl et al. 2019), knitr v. 1.47 (Xie 2014, 2015, 2024), latex2exp v. 0.9.6 (Meschiari 2022), Matrix v. 1.6.5 (Bates, Maechler, and Jagan 2024), MetricGraph v. 1.3.0.9000 (Bolin, Simas, and Wallin 2023b, 2023a, 2023c, 2024; Bolin et al. 2023), patchwork v. 1.2.0 (Pedersen 2024), plotly v. 4.10.4 (Sievert 2020), plotrix v. 3.8.4 (J 2006), reshape2 v. 1.4.4 (Wickham 2007), rmarkdown v. 2.27 (Xie, Allaire, and Grolemund 2018; Xie, Dervieux, and Riederer 2020; Allaire et al. 2024), rSPDE v. 2.3.3.9000 (Bolin and Kirchner 2020; Bolin and Simas 2023; Bolin, Simas, and Xiong 2023), scales v. 1.3.0 (Wickham, Pedersen, and Seidel 2023), sf v. 1.0.16 (Pebesma 2018; Pebesma and Bivand 2023), tidyverse v. 2.0.0 (Wickham et al. 2019), xaringanExtra v. 0.8.0 (Aden-Buie and Warkentin 2024).

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